Infrared Universality: The $r^{-3}$ Spectral Threshold for Coupled Gravitational and Electromagnetic Fields

TL;DR

This paper establishes that an $r^{-3}$ curvature decay rate acts as a universal threshold in asymptotically flat manifolds, separating different spectral behaviors of coupled gravitational and electromagnetic fields and linking to memory effects.

Contribution

It identifies the $r^{-3}$ decay as a fundamental geometric boundary for spectral delocalization and memory phenomena in gauge and gravitational theories.

Findings

Curvature decay faster than $r^{-3}$ leads to essentially self-adjoint operators.

Decay at exactly $r^{-3}$ causes zero modes to become delocalized.

Simulations confirm the predicted quadrupolar and dipolar sky maps for memory fields.

Abstract

We identify the $r^{-3}$ curvature-decay rate as a universal geometric threshold separating compact from non-compact perturbations of Laplace-type operators on asymptotically flat manifolds. For the coupled Einstein--Maxwell system, we prove that the linearized operator $\mathcal{L}$ is essentially self-adjoint and that curvature and field strengths decaying faster than $r^{-3}$ act as relatively compact perturbations, while decay exactly at $r^{-3}$ places $0\in\sigma_{\mathrm{ess}}(\mathcal{L})$ through delocalized zero modes. This threshold mechanism unifies the infrared behavior of spin-1, spin-2, and mixed spin-$(1\oplus2)$ fields, linking the onset of spectral delocalization with the appearance of gravitational and electromagnetic memory. Finite-difference simulations corroborate the analytic scaling and reproduce the characteristic quadrupolar and dipolar sky maps predicted for the coupled memory fields. These results demonstrate that curvature decay at $r^{-3}$ constitutes a fundamental geometric boundary underlying infrared universality in gauge and gravitational theories, providing a spectral counterpart to the asymptotic-symmetry and soft-theorem formulations of memory.